Data Challenge #6
Amanda Carrico
2023-12-09
Github link: https://github.com/agcarrico7/Data-Challenge-6
## load in required libraries
library(hbim)
library(mvtnorm)
library(tidyverse)
library(corrplot)
library(plotly)
library(mclust)
library(knitr)Simulate Data
We will begin by simulating some data to do principal component analysis (PCA) and clustering on. Use the following provided code to simulate data for this exercise. We will be simulating data from two groups.
## set a seed for reproducibility
set.seed(12345)
## create an exhangeable variance covariance for the data
sigma <- make.v(n = 100, r = 0.6, sig2 = 1)
## create centers for the two clusters.
center_1 <- rep(1, 100)
center_2 <- rep(3, 100)
## simulate data for two groups from a multivariate normal
## distribution
data = rbind(rmvnorm(50, mean = center_1, sigma = sigma), rmvnorm(50,
mean = center_2, sigma = sigma))
## add a group label to the data
data = data.frame(group = c(rep(1, 50), rep(2, 50)), data)
head(data)Visualize the Data
Next we will visualize the data.
- Create density plots colored by group membership of the first three variables in the dataset. Comment on what you observe.
# filter data for each group
group_1 <- data %>%
filter(group == 1)
group_2 <- data %>%
filter(group == 2)
# use filtered data to plot density for both groups on same
# plot and color by group, add title and other aesthetics,
# for variable X1
ggplot() + geom_density(mapping = aes(x = group_1$X1, col = "Group 1")) +
geom_density(mapping = aes(x = group_2$X1, col = "Group 2")) +
labs(x = "X1", title = "Density Plot of X1 by Group", col = "Group") +
theme(plot.title = element_text(hjust = 0.5)) + theme_minimal()
# do same for next plot but for X2
ggplot() + geom_density(mapping = aes(x = group_1$X2, col = "Group 1")) +
geom_density(mapping = aes(x = group_2$X2, col = "Group 2")) +
labs(x = "X2", title = "Density Plot of X2 by Group", col = "Group") +
theme(plot.title = element_text(hjust = 0.5)) + theme_minimal()
# again, create same plot but for X3 variable
ggplot() + geom_density(mapping = aes(x = group_1$X3, col = "Group 1")) +
geom_density(mapping = aes(x = group_2$X3, col = "Group 2")) +
labs(x = "X3", title = "Density Plot of X3 by Group", col = "Group") +
theme(plot.title = element_text(hjust = 0.5)) + theme_minimal()
In general, group 2 has higher values for all variables - their median is larger and most values appear to be when compared to group 1. Group 1 also appears to be more right skewed while group 2 appears more left skewed. The shapes of the distributions appear fairly similar for all three variables, with X2 varying the most from X1 and X3. The variables vary from -2 to 6 for the most for each one - it is interesting that all of these variables have a similar range. Scaling may not be necessary but to ensure no impact may anyway.
- Look at the correlation of the data using corrplot::corrplot. Comment on what you observe.
# use cor function to get matrix of correlation
# coefficients for first 3 variables within group 1 filter,
# store matrix in variable
cor_1 <- cor(group_1 %>%
select(X1, X2, X3))
# use stored matrix to make correlation plot
corrplot(cor_1, title = "Correlation Plot of X1, X2, and X3 for Group 1",
addCoef.col = 1, tl.cex = 0.5, number.cex = 0.5, cex.main = 0.5)
# repeat previous steps for the first 3 variables within
# the group 2 filter
cor_2 <- cor(group_2 %>%
select(X1, X2, X3))
corrplot(cor_2, title = "Correlation Plot of X1, X2, and X3 for Group 2",
addCoef.col = 1, tl.cex = 0.5, number.cex = 0.5, cex.main = 0.5)
# repeat again to get correlation plot for no filtered,
# total, data
cor_tot <- cor(data %>%
select(X1, X2, X3))
corrplot(cor_tot, title = "Correlation Plot of X1, X2, and X3",
addCoef.col = 1, tl.cex = 0.5, number.cex = 0.5, cex.main = 0.5)
# show plot for all variables
cor_all <- cor(data)
corrplot(cor_all, title = "Correlation Plot of all Variables in Simulated Data",
addCoef.col = 1, tl.cex = 0.5, cex.main = 0.5, number.cex = 0.5)
Regardless of group, the correlation plots appear fairly similar, with most correlations appearing to be at least .6. Group 2 appeared to have the weakest correlations, closest to .6 and using total data had the most with correlations between variables all being around .8. The variables that were the most highly correlated for the total data as well as by groups were X1 and X3. The variables that were the least correlated were X2 and X3 or X2 and X1. With the range for correlation coefficient being from around .68 to .87 for all variable relationships in all groupings (total data, group 1, and group 2), these variables appear to be strongly linearly correlated with each other despite differing groups.
Perform PCA on the Data
- Perform PCA on the data.
# data has similar variances but want to ensure no effect
# so scale
data_scaled <- scale(data %>%
select(-group))
# use prcomp function for PCA and store for later use
pca_data <- prcomp(data_scaled, center = FALSE, scale. = FALSE)- Make a plot of the cumulative variance explained by the PCs.
# create scree plot of cumulative variation for each PC
ggplot(mapping = aes(x = 1:100, y = summary(pca_data)$importance[3,
])) + geom_line() + labs(title = "Scree Plot of PCA Cumulative Variance Explained",
x = "PC", y = "Cumulative Variance Explained") + theme_minimal() +
theme(plot.title = element_text(hjust = 0.5))
- Make bivariate plots of all combinations of the scores on the first, second, and third PC colored by group membership.
# store scores of PCA in new variable and add the FoodGroup
# variable to tibble for plot later
pca_data_scores <- as_tibble(pca_data$x[, 1:3])
pca_data_scores <- pca_data_scores %>%
mutate(Group = factor(data$group))
# create scatter plot using scores for PC1 vs PC2, PC1 vs
# PC3, and PC2 vs PC3 colored by the Group variable, use
# plotly_build function to make plots interaction
PC1_2 <- ggplot(pca_data_scores, aes(x = PC2, y = PC1, col = Group)) +
geom_point() + labs(title = "PC1 vs PC2 Scores by Group Membership") +
theme(plot.title = element_text(hjust = 0.5))
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## - attr(*, "validate")= logi TRUEplotly_build(PC1_2)PC1_3 <- ggplot(pca_data_scores, aes(x = PC3, y = PC1, col = Group)) +
geom_point() + labs(title = "PC1 vs PC3 Scores by Group Membership") +
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Cluster the original data into 2 clusters using
1. k-means
# use scaled data from PC analysis
# set seed
set.seed(100)
# use kmeans function to cluster and show results
k_means_results <- kmeans(data_scaled, 2, nstart = 50)
# add results of clustering to original data
data_kmean <- data %>%
mutate(kmeans_cluster = k_means_results$cluster)2. Gaussian mixture model
# filter out group
data_used <- data %>%
select(-group)
# use mclust functions to customize models for 2 and
# cluster
BIC <- mclustBIC(data_used)
model <- Mclust(data_used, G = 2, x = BIC)
# add classifications from gaussian mixture model to
# dataframe
data_gaussian <- data %>%
mutate(gaussian_class = model$classification)Create a contingency matrix with the true cluster labels to summarize each clustering result.
# create column for contingency table containing true
# groupings
data_1 <- data %>%
filter(group == 1) %>%
count()
data_2 <- data %>%
filter(group == 2) %>%
count()
num_true <- rbind(data_1, data_2)
names(num_true) <- "True Groupings"
# create column for kmeans clustering
kmeans_1 <- data_kmean %>%
filter(kmeans_cluster == 1) %>%
count()
kmeans_2 <- data_kmean %>%
filter(kmeans_cluster == 2) %>%
count()
num_kmeans <- rbind(kmeans_1, kmeans_2)
names(num_kmeans) <- "Kmeans Groupings"
# also make column for matrix of how many correct kmeans
# groupings
kmeans_correct1 <- data_kmean %>%
filter(kmeans_cluster == 1 & group == 1) %>%
count()
kmeans_correct2 <- data_kmean %>%
filter(kmeans_cluster == 2 & group == 2) %>%
count()
correct_k <- rbind(kmeans_correct1, kmeans_correct2)
names(correct_k) <- c("Correct Kmeans Groupings")
# create column for gaussian mixture model
model_1 <- data_gaussian %>%
filter(gaussian_class == 1) %>%
count()
model_2 <- data_gaussian %>%
filter(gaussian_class == 2) %>%
count()
num_model <- rbind(model_1, model_2)
names(num_model) <- "Gaussian Mixture Model Groups"
# correct groupings
model_correct1 <- data_gaussian %>%
filter(gaussian_class == 1 & group == 1) %>%
count()
model_correct2 <- data_gaussian %>%
filter(gaussian_class == 2 & group == 2) %>%
count()
correct_model <- rbind(model_correct1, model_correct2)
names(correct_model) <- c("Correct Gaussian Mixture Model Groupings")
# combine to make table
contin_table <- cbind(num_true, num_kmeans, num_model)
rownames(contin_table) <- c("Group 1", "Group 2")
kable(contin_table, caption = "Number per Group and Grouping Method")| True Groupings | Kmeans Groupings | Gaussian Mixture Model Groups | |
|---|---|---|---|
| Group 1 | 50 | 46 | 48 |
| Group 2 | 50 | 54 | 52 |
# table for correct groupings
correct_groupings <- cbind(correct_k, correct_model)
rownames(correct_groupings) <- c("Group 1", "Group 2")
kable(correct_groupings, caption = "Number of Correct Groupings per Group and Method")| Correct Kmeans Groupings | Correct Gaussian Mixture Model Groupings | |
|---|---|---|
| Group 1 | 5 | 6 |
| Group 2 | 9 | 8 |
Neither the Gaussian Mixture Model classification nor the kmeans clusters appear to be very accurate.
- Comment on what you observe.
In both the kmeans and the gaussian mixture model, using PC score vectors gave more accurate results, though group 2 for gaussian had an issue. It would appear that kmeans clustering the first 10 principal component score vectors was gave the best results.